Constant-recursive sequence

In mathematics and theoretical computer science, a constant-recursive sequence is an infinite sequence of numbers in which each number in the sequence is equal to a fixed linear combination of one or more of its immediate predecessors. The concept is variously known as a linear recurrence sequence, linear-recursive sequence, linear-recurrent sequence, a C-finite sequence, or a solution to a linear recurrence with constant coefficients.

A prototypical example is the Fibonacci sequence ${\displaystyle 0,1,1,2,3,5,8,13,\ldots }$, in which each number is the sum of the previous two. The power of two sequence ${\displaystyle 1,2,4,8,16,\ldots }$ is also constant-recursive because each number is the sum of twice the previous number. The square number sequence ${\displaystyle 0,1,4,9,16,25,\ldots }$ is also constant-recursive. However, not all sequences are constant-recursive; for example, the factorial sequence ${\displaystyle 1,1,2,6,24,120,\ldots }$ is not constant-recursive. All arithmetic progressions, all geometric progressions, and all polynomials are constant-recursive.

Formally, a sequence of numbers ${\displaystyle s_{0},s_{1},s_{2},s_{3},\ldots }$ is constant-recursive if it satisfies a recurrence relation

$${\displaystyle s_{n}=c_{1}s_{n-1}+c_{2}s_{n-2}+\dots +c_{d}s_{n-d},}$$

where ${\displaystyle c_{i}}$ are constants. For example, the Fibonacci sequence satisfies the recurrence relation ${\displaystyle F_{n}=F_{n-1}+F_{n-2},}$ where ${\displaystyle F_{n}}$ is the ${\displaystyle n}$th Fibonacci number.

Constant-recursive sequences are studied in combinatorics and the theory of finite differences. They also arise in algebraic number theory, due to the relation of the sequence to polynomial roots; in the analysis of algorithms, as the running time of simple recursive functions; and in the theory of formal languages, where they count strings up to a given length in a regular language. Constant-recursive sequences are closed under important mathematical operations such as term-wise addition, term-wise multiplication, and Cauchy product.

The Skolem–Mahler–Lech theorem states that the zeros of a constant-recursive sequence have a regularly repeating (eventually periodic) form. On the other hand, the Skolem problem, which asks for an algorithm to determine whether a linear recurrence has at least one zero, remains unsolved.